1. ## Identities Applications

Hello,

I'm having trouble with a trigonometry problem, outlined below:

cos²(2x) - sin²(2x) = (√3/2)

I'm not quire sure how to solve for x, can any of you help?

Thanks

2. Recall the identity cos²x-sin²x = cos(2x).

3. I still don't quite understand the relationship between the identity you suggested:

cos²x - sin²x = cos(2x)

and the left term in the equation I have:

cos²(2x) - sin²(2x)

I'm not quite sure of what to do because of the '2x', otherwise I could easily substitute the identity you suggested.

EDIT:

I see now that in my case it would be cos(4x) as the substitution.

4. Okay, I've determined that the following is true:

cos(4x) = (√3/2)

How do I simplify further?

5. Originally Posted by ElectroNerd
I see now that in my case it would be cos(4x) as the substitution.
Yes, and (√3/2) = cos(π/6). So you've cos(4x) = cos(π/6), which implies 4x = π/6. Can you find the other solutions from that?

6. Okay, I didn't realize that other identity!

So I solved for x:

x = π/24

Just so I can understand how to solve more problems, how do you go about this? Do you have all the identities memorized? If I knew that cos(4x) = cos(π/6), then I could have solved this more easily.

I guess I should just memorize all the identities, is that the best way?

7. Originally Posted by ElectroNerd
So I solved for x: x = π/24
Right! That's what one solution, though. There are infinitely many. Can you find (or are you required to) find the rest?
Just so I can understand how to solve more problems, how do you go about this? Do you have all the identities memorized? If I knew that cos(4x) = cos(π/6), then I could have solved this more easily.
No, cos(4x) = cos(π/6) is not an identity. Our equation was cos(4x) = (√3/2), but (√3/2) = cos(π/6) so we have cos(4x) = cos(π/6).

I guess I should just memorize all the identities, is that the best way?
Memorisation will only get you so far; the best way is doing as many problems as you can get hold of.

8. That is the answer they give, thanks!

Do you know of any good trigonometry references I can study?

9. Originally Posted by ElectroNerd
Do you know of any good trigonometry references I can study?
A brilliant book is this one: Advanced Trigonometry - C. V. Durell and A. Robson. It's very old, so you can probably download it online for free too.